The inclusive production of open heavy-flavour hadrons in hadronic collisions is computed using the collinear factorisation approach [1]} as a convolution
of the parton distribution functions of the incoming hadrons, the hard parton–parton scattering cross section computed perturbatively, and the fragmentation process describing the non-perturbative evolution of a charm- or beauty-quark into an open heavy-flavour hadron.
These calculations are implemented at the next-to-leading order (NLO) accuracy in the general-mass variable-flavour-number scheme
(GM-VFNS) [2]}, [3]}, and at NLO with an all-order resummation to next-to-leading log (NLL) accuracy in the limit where the \(p_{\rm T}\)  of the heavy quark is much larger than its mass in the FONLL resummation approach [4]}, [5]}.
Recent calculations with next-to-next-to-leading-order (NNLO) QCD radiative corrections are implemented for the beauty-quark production cross section [6]}.
Other predictions are also performed in the leading order (LO) approximation through the \(k_{\rm T}\) -factorisation framework [7]}.
All these computations describe, albeit within large theoretical uncertainties, the production cross sections of open heavy-flavour hadrons measured in pp and p\(\overline{\rm p}\)  collisions
in different kinematic domains at centre-of-mass energies ranging from 0.2 to 13 TeV [8]}, [9]}, [10]}.
Non-prompt \(\rm J/\psi \)  production is directly related to open beauty-hadron production, and can be used to estimate the latter after an extrapolation.
The measurements of the total beauty-quark production cross sections are less sensitive to the non-perturbative hadronisation effects than the total charm–quark production, which makes them a good test for QCD in the perturbative regime.
In the context of the LHC heavy-ion physics programme, the measurement of beauty production in pp collisions is crucial for studying both cold and hot nuclear matter effects, as they provide a reference for the beauty-hadron production measurements in proton–nucleus and nucleus–nucleus collisions.
